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Parametric Equations | Functions Math Help

How can we use the concept:

Parametric equations introduce a third variable t for the parameter and use this to define x and y. The variable t is used because it often specifies the interval of time in these equations. You can then use this to help define the equation. For example, you can say at t = “amount” the position of “object” is “ordered pair”.

When working with equations, you have used a process to sketch functions where you:

Choose the x value
Use a function f to find y i.e. y = f(x)
Plot the point (x,y)

You have often used a table of values for this process.

The same can be done with parametric equations, but you need to add that t value for the parameter of the shape (again, a curved shape). We will discuss this strategy further in our solved examples. We will also look at a strategy where we remove the parameter.

The graph of parametric equations can cross over itself which can be a question seen in problem sets.

You can also describe parametric equations with boundaries such as studying when an object would hit the ground - the graph of that parametric equation would continue beyond x = 0 but you would have no interest in that data because your object has hit the ground.

If you are asked questions regarding boundaries such as t >2. This would affect the values in your table - you would only choose values greater than 2. You can also see boundaries such as -3 < t < 1which would again, dictate which values you can use for t in your table.

Some tips:

I have a graphing calculator ready and plot my points as I go.
If I feel I have missing data, I can add points into the table to gain more information.

Sample Math Problems

Question

Sketch the parametric curve for the following equations.

x =

t^2

+ t

y = 2t - 1

Answer

Strategy 1: Using a table of values to sketch the graph.

You are going to use a table to choose t values and then solve. There isn’t a strict rule on how many t values to use but you must use enough to get a good idea of the sketch of the curve.

At this point I graph my points and sketch my curve.

You will notice from here that you can gain basic knowledge of the curve but the points I chose didn’t quite define the vertex, so I can always go back to my table and add a few more values for t.

This sketch provides us with more information regarding the vertex.

Strategy 2: Eliminate the parameter.

x =

t^2

+ t

y = 2t - 1

For this strategy we need to solve one of the equations for t and then substitute that into the other equation.

In this case, solving for t in y equation is not very difficult.

y = 2t -1

y + 1 = 2t move the constant using inverse operations

\frac{1}{2}

(y + 1) = t divide by 2 on each side

Now we can plug this into the x equation.

x =

t^2

+ t

x = (

\frac{1}{2}

(y + 1))

^2

+ (

\frac{1}{2}

(y + 1)) Simplify this equation.

x =

\frac{1}{4}y^2

+ y +

\frac{3}{4}

From here, to sketch a graph you would use an xy table and substitute values in to get your ordered pairs.

Question

Sketch a graph of the following parametric equations.

x = 4 -

2t^2

y = 2 + 4t

Answer

- I am using a table for this example.

Step 1: Choose t values.

Step 2: substitute into your equations to get your x and y values.

Step 3: plot your ordered pairs on a graph and sketch your curve.

If you are asked questions regarding boundaries such as t >2. This would affect the values in your table - you would only choose values greater than 2.

Question

Sketch the parametric curve for the following equations.

x = t

y =